A coloring of a digraph is partition its vertex set such that each class induces with no directed cycles. k k -chromatic if ...

This study suggests that the smallest physical constant exists for product of space interval, time and energy. The integer times which correspond to energy, interval any particle. Combined with fractal geometry, many parameters cannot be calculated using current theory can through our new method. As a theoretical prediction, these results will tested by experiment.

Let χ be a non-principal Dirichlet character modulo a prime p. Let q1 < q2 denote the two smallest prime non-residues of χ. We give explicit upper bounds on q2 that improve upon all known results. We also provide a good upper estimate on the product q1q2 which has an upcoming application to the study of norm-Euclidean Galois fields.

We calculate the kinetic-decoupling temperature for weakly interacting massive particles (WIMPs) in supersymmetric (SUSY) and extra dimensional models that can account for the cold-dark-matter abundance determined from cosmic microwave background measurements. Depending on the parameters of the particle-physics model, a wide variety of decoupling temperatures is possible, ranging from several M...